Representation theory in chiral conformal field theory: from fields to observables

Abstract

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and the representation theory of vertex operator algebras. We define a correspondence between representations in the two contexts, and show how to construct representations of conformal nets from VOAs. We also show that this correspondence is rich enough to relate the respective 'fusion product' theories for conformal nets and VOAs, by constructing local intertwiners (in the sense of conformal nets) from intertwining operators (in the sense of VOAs). We use these techniques to show that all WZW conformal nets can be constructed using our geometric interpolation procedure.

79 pages. v2: minor revisions and updates

References in corpus (3)

• W-algebras as coset vertex algebras
• Structure and Classification of Superconformal Nets
• Unitarity of the modular tensor categories associated to unitary vertex operator algebras, I

Cited by in corpus (14)

• Subfactors and Mathematical Physics
• Q-systems and extensions of completely unitary vertex operator algebras
• Fusion and positivity in chiral conformal field theory
• Convergence of Sewing Conformal Blocks
• Wightman fields for two-dimensional conformal field theories with pointed representation category
• Energy bounds for vertex operator algebra extensions
• From vertex operator superalgebras to graded-local conformal nets and back
• Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
• Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules
• Non-unitary Wightman CFTs and non-unitary vertex algebras
• Regular vertex operator subalgebras and compressions of intertwining operators
• Unitary vertex algebras and Wightman conformal field theories
• Polynomial energy bounds for type $F_4$ WZW-models
• Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case